※ [本文轉錄自 NTU-Exam 看板]
作者: jhs0909 (1213視小物治之夜) 看板: NTU-Exam
標題: [試題] 94上 陳其誠 微積分乙 第二次考試
時間: Tue Nov 29 14:59:40 2005
課程名稱︰微積分乙
課程性質︰共同必修
課程教師︰陳其誠
開課系所︰醫學院各系、生科、農化、公衛等系
考試時間︰2005.11.29
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning. This exam contains eight problems
written in one page.
(1) (5 points each) Cauculate the following limits:
(a)lim (e^2x -1)/x (b)lim (11x^3 +29x^2 +2005)/(x^3 +x-1)
x→0 x→∞
(c)lim (lnx)^100 /x (d)lim (1+cosx)/sinx
x→∞ x→π
(2) For each of the following functions, find the n'th Taylor polynomial
Pn(x) at x=α:
(a) f(x)= lnx, α=1, n=3 (7 points)
(b) f(x)= sinx, α=π, n=10 (8 points)
(c) f(x)= e^x, α=0, n=1 (10 points)
(3) (10 points) A person uses Newton's method to solve the equation x^3 +x+1=0
by taking x0= -1/2. Calculate x1.
(4) (10 points) Find the maximum of the function P(t)= 1000+ 45/(25+ t^2) for
t≧0
(5) (10 points each) For the following dynamical systems, find every equilib-
rium and determine the stability of each equilibrium: (t及t+1為下標)
(a) M(t+1) = Mt- 0.5Mt/(1+0.1Mt) +1
(b) b(t+1) = 2(1- bt/500)bt
(6) (5 points) The dynamical system x(t+1) = 3(1-xt)xt has a positive equilib-
rium. Is it stable? Why?
(7) (5 points) Suppose that f(x) is a differentiable function on (-∞, ∞) and
the equation f'(x)=0 has less than 20 distinct roots. Is it possible for
the equation f(x)=0 to have more than 20 distinct roots? Why?
(8) (5 points) Is it possible that a discrete dynamical system has more than
one (locally) stable equilibrium? If you say yes, give an example; if you
say no, give a reason.
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